Calculations of the structure of basin volumes for mechanically stable packings

Abstract

Experimental and computational model systems composed of frictionless particles in a fixed geometry have a finite number of distinct mechanically stable (MS) packings. The frequency of occurrence for each MS packing is highly variable and depends strongly on preparation protocol. Despite intense work, it is extremely difficult to predict a priori the MS packing probabilities. We describe a novel computational method for calculating the volume and other geometrical properties of the 'basin of attraction' for each MS packing. The basin of attraction for a MS packing contains all initial conditions in configuration space that map to that MS packing using a given preparation protocol. We find that the basin is a highly complex structure. For a compressive-quench-from-zero-density protocol, we show the existence of a small core volume of the basin around each MS packing for which all points map to that MS packing. However, in contrast to previous studies for supercooled liquids, glasses, and over-compressed jammed systems, we find that the MS packing probabilities are very weakly correlated with this core volume. Instead, MS packing probabilities obtained from compression protocols that use initially dilute configurations and do not allow particle overlaps (i.e. those relevant to granular media) are determined by complex geometric features of the basin of attraction that are distant from the MS packing. In particular, we find that the shape of the average basin profile function S(l), which gives the probability for a point on a hyper-spherical shell a distance l from a given MS packing to map back to that packing, can be described by a Γ-distribution with a peak that increases as the system size increases and as the quench rate decreases. We find a simple model which predicts S(l) for the extreme cases of very slow and fast quench rates.

Protocol dependence of the jamming transition

“…3 (c)). We imagine that a one-dimensional trajectory L(φ, γ) through configuration space will encounter the basin of attraction [19] of a jammed packing with a probability F (φ)S(φ)dL during a step of size dL in configuration space, where S(φ) is the 2N − 1- dimensional cross-section of the basin of attraction of a jammed packing perpendicular to dL.…”

Jammed disks in a narrow channel: criticality and ordering tendencies

“…The methodology as presented here is amenable to further development in several directions: (i) the regime 1 + 3/4 < H/σ < 2 of 32 tiles as identified in [14], (ii) mixtures of disks with different radii and masses including rattlers [35,36,37], (iii) spatial correlations of heterogeneities in mass density, and (iv) channels with the axis oriented vertically.…”

Structural analysis of high-dimensional basins of attraction

“…However, the approach of ref. [25] breaks down for higher dimensional systems for which most of the volume of the basin is concentrated at distances from the 'minimum' where the overwhelming majority of points do not belong to the basin. The method that we present here allows us to explore precisely those very rarified regions where most of the 'mass' of a basin is concentrated.…”

“…Ashwin et al [25], defined the basin of attraction as the collection of initial zero-density configurations that evolve to a given jammed packing of soft repulsive disks via a compressive quench. On the basis of 'brute-force' calculations on low-dimensional systems, Ashwin et al suggested that basins of attraction tend to be "branched and threadlike" away from a spherical core region.…”

“…Hence, using a Monte Carlo 'rejection method' to compute the volume of the non-convex body as the fraction of volume contained in a hypersphere [24,25], does not yield accurate results: the largest contribution should come from points that are barely sampled, if at all.…”